This book contains a selection of advanced topics suitable for final year undergraduates in science and engineering, and is based on courses of lectures given by one of us (G. S.) to various groups of third year engineering and science students at Imperial College over the past 15 years. It is assumed that the student has a good understanding of basic ancillary mathematics. The emphasis in the text is principally on the analytical understanding of the topics which is a vital prerequisite to any subsequent numerical and computational work. In no sense does the book pretend to be a comprehensive or highly rigorous account, but rather attempts to provide an accessible working knowledge of some of the current important analytical tools required in modern physics and engineering. The text may also provide a useful revision and reference guide for postgraduates.

Each chapter concludes with a selection of problems to be worked, some of which have been taken from Imperial College examination papers over the last ten years. Answers are given at the end of the book.

We wish to thank Dr Tony Dowson and Dr Noel Baker for reading the manuscript and making a number of helpful suggestions.

In Chapter 1 we introduced some particular functions of a complex variable, such as powers and the exponential, that were needed for the chapters to follow. We now take up the subject again and develop a general theory of functions of a complex variable, including integration in the complex plane. Since the plane has no physical meaning it might seem that we are embarking on a study that has no relevance to an engineer, but a familiarity with complex-variable techniques is important in all branches of engineering. In particular, they are essential for a proper appreciation of the Fourier- and inverse Laplace-transform operations discussed in later chapters, and this is the reason for their consideration now.

General properties

We consider complex functions of the complex variable z = x + iy, where the definition of a function is analogous to that in real variable theory:

A domain is simply an open region of the complex plane bounded by a closed contour, and we shall use such terms with no more precise definition than geometrical intuition requires.

Having now arrived at the heart of our subject, we show how the mathematical techniques that have been discussed can be used to solve some of the problems associated with linear systems.

Basic concepts

In its most general form a system is any physical device that when stimulated or excited, produces a response. The system could be as complicated as (a model of) the human brain or as simple as an electrical circuit with a lumped resistance, and the problem the engineer could face is to determine the response or output of the system resulting from a known excitation or input. To develop procedures that are applicable to a variety of systems, it is necessary to restrict the type of system considered.

Physical description

For the systems of concern to us it is assumed that the input and output are functions of a single real variable t, which we shall speak of as time, and that a causal relationship exists between the two. The term causal implies that the output is a function of the input alone, and because of this, the system is “nonanticipatory” that is, there is no output until the input is applied. This is an attribute of all physically realizable systems. For convenience we shall refer to the input and output as signals, and if these are denoted by f(t) and x(t), respectively, a system can be depicted as shown in Fig. 3.1.

For the past decade and more, all students in electrical and computer engineering at the University of Michigan have been required to take a course concerned with the mathematical methods for the solution of linear-systems problems. The course is typically taken in the junior year after completion of a basic four-semester mathematics sequence covering analytic geometry, matrices and determinants, differential and integral calculus, elementary differential equations, and so forth. Because it is the last course in mathematics that all must take for a bachelor's degree, a number of topics are included in addition to those customary in an introductory systems course, for example, functions of a complex variable with particular reference to integration in the complex plane, and Fourier series and transforms.

Some of the courses that make use of this material can be taken concurrently thereby constraining the order in which the topics are covered. It is, for example, necessary that Laplace transforms be introduced early and the treatment carried to such a stage that the student is able to use the transform to solve initial-value problems. Fourier series must also be covered in the first half of the course, and the net result is an ordering that is different from the mathematically natural one, but that is quite advantageous in practice. Thus, the relatively simple material dealing with the Laplace transform and its applications comes at the beginning and provides the student with a sense of achievement prior to the introduction of the more abstract material on functions of a complex variable.